Modeling with uncertainty: A Bayesian parameter estimation tutorial
Introduction
Technical objective
Learn the basics of Bayesian parameter estimation and how it differs from frequentist parameter estimation
Substantive research question
How do emotion regulation strategies moderate the association between sleep and negative emotions?
Introductory remarks
In this tutorial, we will be applying the Bayesian lens to the estimation of parameters in a multilevel linear model. In particular, we will examine two emotion regulation strategies, cognitive reappraisal and expressive suppression, and whether they moderate the association between hours of sleep at night and negative affect the following day.
Briefly, in cognitive reappraisal, one changes the way in which they think about an emotionally-inducing event in order to alter their emotional response to that event. For example, instead of thinking about an exam as a difficult and consequential test of their abilities, a student might choose to frame the exam as an opportunity to show what they have learned. In expressive suppression, one regulates their emotions by inhibiting their external displays of emotion. For example, a student who feels nervous about an upcoming exam may stop themselves from showing a worried look on their face and telling their friends that they feel nervous. More information on these two emotion regulation strategies can be found in Gross & John (2003).
TODO: add more about the participants of the study, the data, and previous literature about the link between emotion regulation/sleep and negative affect/affective regulation
TODO: add section headers throughout (using # and ##, etc.) so that they show up under robobook ToC TODO: add commentary throughout to guide reader through the steps of the tutorial TODO: use this link for some example plots/analysis of Bayesian MLMs: https://www.rensvandeschoot.com/tutorials/brms-started/
Preliminaries
# set filepath for data file
filepath <- "https://raw.githubusercontent.com/LRI-2/Data/main/ILD/AMIBshare_persons_2019_0501.csv"
# read in the .csv file using the url() function
AMIB_persons <- read.csv(file=url(filepath),header=TRUE)Data manipulation
# subset to day-level variables of interest
bpe_daily <- AMIB_daily[,c("id", "day", "slphrs", "negaff")]
# subset to person-level variables of interest
bpe_persons <- AMIB_persons[,c("id", "erq_reap", "erq_supp")]
# merge day- and person-level data
bpe_data <- merge(bpe_daily, bpe_persons, by = "id")Initial data plotting
Cognitive reappraisal model
bpe.reap <- brm(negaff ~ 1 + day + slphrs + erq_reap_c + slphrs:erq_reap_c + (1 + slphrs + erq_reap_c|id), data = bpe_data, family = gaussian(),
iter = 2000, chains = 4, cores = 4)## Warning: Rows containing NAs were excluded from the model.## Compiling Stan program...## Trying to compile a simple C file## Running /Library/Frameworks/R.framework/Resources/bin/R CMD SHLIB foo.c
## using C compiler: ‘Apple clang version 15.0.0 (clang-1500.0.40.1)’
## using SDK: ‘’
## clang -arch arm64 -I"/Library/Frameworks/R.framework/Resources/include" -DNDEBUG -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/Rcpp/include/" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppEigen/include/" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppEigen/include/unsupported" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/BH/include" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/StanHeaders/include/src/" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/StanHeaders/include/" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppParallel/include/" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/rstan/include" -DEIGEN_NO_DEBUG -DBOOST_DISABLE_ASSERTS -DBOOST_PENDING_INTEGER_LOG2_HPP -DSTAN_THREADS -DUSE_STANC3 -DSTRICT_R_HEADERS -DBOOST_PHOENIX_NO_VARIADIC_EXPRESSION -DBOOST_NO_AUTO_PTR -include '/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/StanHeaders/include/stan/math/prim/fun/Eigen.hpp' -D_REENTRANT -DRCPP_PARALLEL_USE_TBB=1 -I/opt/R/arm64/include -fPIC -falign-functions=64 -Wall -g -O2 -c foo.c -o foo.o
## In file included from <built-in>:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/StanHeaders/include/stan/math/prim/fun/Eigen.hpp:22:
## In file included from /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppEigen/include/Eigen/Dense:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppEigen/include/Eigen/Core:88:
## /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:628:1: error: unknown type name 'namespace'
## namespace Eigen {
## ^
## /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:628:16: error: expected ';' after top level declarator
## namespace Eigen {
## ^
## ;
## In file included from <built-in>:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/StanHeaders/include/stan/math/prim/fun/Eigen.hpp:22:
## In file included from /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppEigen/include/Eigen/Dense:1:
## /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppEigen/include/Eigen/Core:96:10: fatal error: 'complex' file not found
## #include <complex>
## ^~~~~~~~~
## 3 errors generated.
## make: *** [foo.o] Error 1## Start sampling## Warning: There were 2 divergent transitions after warmup. See
## https://mc-stan.org/misc/warnings.html#divergent-transitions-after-warmup
## to find out why this is a problem and how to eliminate them.## Warning: Examine the pairs() plot to diagnose sampling problems## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## https://mc-stan.org/misc/warnings.html#bulk-ess## Warning: There were 2 divergent transitions after warmup. Increasing
## adapt_delta above 0.8 may help. See
## http://mc-stan.org/misc/warnings.html#divergent-transitions-after-warmup## Family: gaussian
## Links: mu = identity; sigma = identity
## Formula: negaff ~ 1 + day + slphrs + erq_reap_c + slphrs:erq_reap_c + (1 + slphrs + erq_reap_c | id)
## Data: bpe_data (Number of observations: 1421)
## Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
## total post-warmup draws = 4000
##
## Group-Level Effects:
## ~id (Number of levels: 190)
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS
## sd(Intercept) 0.98 0.13 0.74 1.23 1.00 1273
## sd(slphrs) 0.10 0.02 0.05 0.14 1.01 393
## sd(erq_reap_c) 0.09 0.07 0.00 0.25 1.00 569
## cor(Intercept,slphrs) -0.75 0.09 -0.88 -0.54 1.00 1863
## cor(Intercept,erq_reap_c) 0.03 0.47 -0.85 0.84 1.00 2146
## cor(slphrs,erq_reap_c) -0.13 0.50 -0.92 0.84 1.00 1612
## Tail_ESS
## sd(Intercept) 1610
## sd(slphrs) 572
## sd(erq_reap_c) 759
## cor(Intercept,slphrs) 2093
## cor(Intercept,erq_reap_c) 2644
## cor(slphrs,erq_reap_c) 2976
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Intercept 3.24 0.13 2.99 3.50 1.00 3979 3574
## day -0.07 0.01 -0.08 -0.05 1.00 8691 2709
## slphrs -0.08 0.02 -0.11 -0.05 1.00 3859 2696
## erq_reap_c -0.01 0.12 -0.25 0.23 1.00 3209 2840
## slphrs:erq_reap_c 0.00 0.02 -0.03 0.03 1.00 3265 2852
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 0.78 0.02 0.75 0.81 1.00 3068 2965
##
## Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).## Estimate Est.Error Q2.5 Q97.5
## R2 0.4452904 0.01828767 0.4076417 0.4791197
TODO: interpret model results (including Bayesian significance testing) TODO: make a visualization of this model that shows the prototypical individual and all individuals
Expressive suppression model
bpe.supp <- brm(negaff ~ 1 + day + slphrs + erq_supp_c + slphrs:erq_supp_c + (1 + slphrs + erq_supp_c|id), data = bpe_data, family = gaussian(),
iter = 2000, chains = 4, cores = 4)## Warning: Rows containing NAs were excluded from the model.## Compiling Stan program...## Trying to compile a simple C file## Running /Library/Frameworks/R.framework/Resources/bin/R CMD SHLIB foo.c
## using C compiler: ‘Apple clang version 15.0.0 (clang-1500.0.40.1)’
## using SDK: ‘’
## clang -arch arm64 -I"/Library/Frameworks/R.framework/Resources/include" -DNDEBUG -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/Rcpp/include/" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppEigen/include/" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppEigen/include/unsupported" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/BH/include" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/StanHeaders/include/src/" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/StanHeaders/include/" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppParallel/include/" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/rstan/include" -DEIGEN_NO_DEBUG -DBOOST_DISABLE_ASSERTS -DBOOST_PENDING_INTEGER_LOG2_HPP -DSTAN_THREADS -DUSE_STANC3 -DSTRICT_R_HEADERS -DBOOST_PHOENIX_NO_VARIADIC_EXPRESSION -DBOOST_NO_AUTO_PTR -include '/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/StanHeaders/include/stan/math/prim/fun/Eigen.hpp' -D_REENTRANT -DRCPP_PARALLEL_USE_TBB=1 -I/opt/R/arm64/include -fPIC -falign-functions=64 -Wall -g -O2 -c foo.c -o foo.o
## In file included from <built-in>:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/StanHeaders/include/stan/math/prim/fun/Eigen.hpp:22:
## In file included from /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppEigen/include/Eigen/Dense:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppEigen/include/Eigen/Core:88:
## /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:628:1: error: unknown type name 'namespace'
## namespace Eigen {
## ^
## /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:628:16: error: expected ';' after top level declarator
## namespace Eigen {
## ^
## ;
## In file included from <built-in>:1:
## In file included from /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/StanHeaders/include/stan/math/prim/fun/Eigen.hpp:22:
## In file included from /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppEigen/include/Eigen/Dense:1:
## /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppEigen/include/Eigen/Core:96:10: fatal error: 'complex' file not found
## #include <complex>
## ^~~~~~~~~
## 3 errors generated.
## make: *** [foo.o] Error 1## Start sampling## Warning: There were 3 divergent transitions after warmup. See
## https://mc-stan.org/misc/warnings.html#divergent-transitions-after-warmup
## to find out why this is a problem and how to eliminate them.## Warning: Examine the pairs() plot to diagnose sampling problems## Warning: There were 3 divergent transitions after warmup. Increasing
## adapt_delta above 0.8 may help. See
## http://mc-stan.org/misc/warnings.html#divergent-transitions-after-warmup## Family: gaussian
## Links: mu = identity; sigma = identity
## Formula: negaff ~ 1 + day + slphrs + erq_supp_c + slphrs:erq_supp_c + (1 + slphrs + erq_supp_c | id)
## Data: bpe_data (Number of observations: 1421)
## Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
## total post-warmup draws = 4000
##
## Group-Level Effects:
## ~id (Number of levels: 190)
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS
## sd(Intercept) 0.94 0.13 0.69 1.20 1.00 1281
## sd(slphrs) 0.09 0.02 0.05 0.13 1.00 447
## sd(erq_supp_c) 0.11 0.07 0.01 0.26 1.00 422
## cor(Intercept,slphrs) -0.73 0.11 -0.87 -0.47 1.00 1632
## cor(Intercept,erq_supp_c) 0.35 0.40 -0.60 0.93 1.00 1406
## cor(slphrs,erq_supp_c) -0.18 0.47 -0.92 0.79 1.00 1006
## Tail_ESS
## sd(Intercept) 1436
## sd(slphrs) 498
## sd(erq_supp_c) 872
## cor(Intercept,slphrs) 1276
## cor(Intercept,erq_supp_c) 2308
## cor(slphrs,erq_supp_c) 1590
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Intercept 3.24 0.13 2.97 3.50 1.00 3824 3168
## day -0.07 0.01 -0.08 -0.05 1.00 8866 3104
## slphrs -0.08 0.02 -0.11 -0.05 1.00 4511 2918
## erq_supp_c 0.09 0.10 -0.11 0.29 1.00 2989 3209
## slphrs:erq_supp_c -0.01 0.01 -0.03 0.01 1.00 3405 3145
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 0.78 0.02 0.75 0.81 1.00 3285 3070
##
## Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).## Estimate Est.Error Q2.5 Q97.5
## R2 0.4439703 0.01782485 0.4068412 0.4779817
TODO: interpret model results (including Bayesian significance testing) TODO: make a visualization of this model that shows the prototypical individual and all individuals
TODO: compare the two models (reap vs. supp - which one seems more predictive of better control of negaff?)
TODO: write up a final analysis/conclusion for this tutorial
Conclusion
[TODO]
References
Gross, J. J., & John, O. P. (2003). Individual differences in two emotion regulation processes: Implications for affect, relationships, and well-being. Journal of Personality and Social Psychology, 85(2), 348–362. https://doi.org/10.1037/0022-3514.85.2.348